臺灣博碩士論文加值系統

We present an ab initio numerical method to study the properties of Bose-Einstein condensates (BECs) which include both interactions via contact and magnetic dipole-dipole forces. We efficiently solve the nonlocal and anisotropic interaction potential between dipoles which are represented in a differential form. The BEC Hamiltonian is discretized and solved accurately through the generalized pseudospectral method (GPS method). Using the iteration minimization technique, we obtain the solutions of the non-linear Gross-Pitaevskii equation with non-local dipolar interaction. We find that the density profiles strongly depend upon the geometry of trapping potentials. We determine that the maximum density is not always located at the center of a trap due to the interaction between dipoles. Experiments have shown that the stability of dipolar BECs strongly depends on the geometry of trapping potentials and the scattering length. As the scattering length decreases under certain critical values acrit , BECs are no longer stable. Using the GPS method, the critical scattering length corresponding to different trap geometries is accurately determined with a minimum number of grid points. In addition, we show that the
Thomas-Fermi approximation is not good enough to describe condensates before BECs collapse, and the double-peaks feature of density profiles is an important characteristic in such condition. In the near future, the dynamics of dipolar BECs will be studied employing the GPS method.

Contents
Acknowledgement i
Abstract (Chinese) ii
Abstract iii
1 Introduction 1
1.1 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Dipolar BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Theory of Bose-Einstein condensate and dipole gas 6
2.1 Effective interactions and the scattering length . . . . . . . . . . . . . . . 6
2.2 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Stability and collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Dipolar interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
iv
3 Numerical methods 15
3.1 Challenge of dipole interaction . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Generalized pseudospectral method . . . . . . . . . . . . . . . . . . . . 16
3.3 BEC in an anisotropic trap . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Dipole-dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Ground state and energy minimization by iterative method . . . . . . . . 24
4 Results and discussions 28
4.1 Pure contact interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Pure dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Scattering length and stability . . . . . . . . . . . . . . . . . . . . . . . 37
5 Conclusions and Perspectives 41
Bibliography 43

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